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Beliebt Algebra >

vereinfachen-0.14log_{2}(0.14)+(-0.86)log_{2}(0.86)

Lösung

vereinfachen

- 0.14 lo g_{2} (0.14) + (- 0.86) lo g_{2} (0.86)

Lösung

- lo g_{2} (\frac{4 3 ^{0.86} \cdot 7 ^{0.14}}{25}) + 1

+1

Dezimale

0.58423 \dots

Schritte zur Lösung

- 0.14 lo g_{2} (0.14) + (- 0.86) lo g_{2} (0.86)

- 0.14 lo g_{2} (0.14) = - 0.14 lo g_{2} (\frac{7}{50})

lo g_{2} (0.86) = lo g_{2} (\frac{43}{50})

= - 0.14 lo g_{2} (\frac{7}{50}) - 0.86 lo g_{2} (\frac{43}{50})

- 0.14 lo g_{2} (\frac{7}{50}) = - lo g_{2} ((\frac{7}{50})^{0.14})

- 0.86 lo g_{2} (\frac{43}{50}) = - lo g_{2} ((\frac{43}{50})^{0.86})

= - lo g_{2} ((\frac{7}{50})^{0.14}) - lo g_{2} ((\frac{43}{50})^{0.86})

Wende die log Regel an:

lo g_{a} (x) + lo g_{a} (y) = lo g_{a} (x y)

- lo g_{2} ((\frac{43}{50})^{0.86}) - lo g_{2} ((\frac{7}{50})^{0.14}) = lo g_{2} ((\frac{43}{50})^{0.86} (\frac{7}{50})^{0.14})

= - lo g_{2} ((\frac{43}{50})^{0.86} (\frac{7}{50})^{0.14})

Vereinfache

(\frac{43}{50})^{0.86} (\frac{7}{50})^{0.14} : \frac{4 3 ^{0.86} \cdot 7 ^{0.14}}{50}

= - lo g_{2} (\frac{4 3 ^{0.86} \cdot 7 ^{0.14}}{50})

Wende die log Regel an:

lo g_{a} (\frac{x}{y}) = lo g_{a} (x) - lo g_{a} (y)

lo g_{2} (\frac{4 3 ^{0.86} \cdot 7 ^{0.14}}{50}) = lo g_{2} (4 3^{0.86} \cdot 7^{0.14}) - lo g_{2} (50)

= - (lo g_{2} (4 3^{0.86} \cdot 7^{0.14}) - lo g_{2} (50))

lo g_{2} (4 3^{0.86} \cdot 7^{0.14}) = 0.86 lo g_{2} (43) + 0.14 lo g_{2} (7)

= - ((0.86 lo g_{2} (43) + 0.14 lo g_{2} (7)) - lo g_{2} (50))

Apply rule:

(a) = a

(0.86 lo g_{2} (43) + 0.14 lo g_{2} (7)) = 0.86 lo g_{2} (43) + 0.14 lo g_{2} (7)

= - (0.86 lo g_{2} (43) + 0.14 lo g_{2} (7) - lo g_{2} (50))

lo g_{2} (50) = 1 + 2 lo g_{2} (5)

= - (0.86 lo g_{2} (43) + 0.14 lo g_{2} (7) - (1 + 2 lo g_{2} (5)))

Wende das Distributivgesetz an:

- (a + b) = - a - b

- (1 + 2 lo g_{2} (5)) = - 1 - 2 lo g_{2} (5)

= - (0.86 lo g_{2} (43) + 0.14 lo g_{2} (7) - 1 - 2 lo g_{2} (5))

0.86 lo g_{2} (43) = lo g_{2} (4 3^{0.86})

0.14 lo g_{2} (7) = lo g_{2} (7^{0.14})

- 2 lo g_{2} (5) = - lo g_{2} (5^{2})

= - (lo g_{2} (4 3^{0.86}) + lo g_{2} (7^{0.14}) - 1 - lo g_{2} (5^{2}))

lo g_{2} (4 3^{0.86}) + lo g_{2} (7^{0.14}) - lo g_{2} (5^{2}) = lo g_{2} (\frac{4 3 ^{0.86} \cdot 7 ^{0.14}}{25})

= - (lo g_{2} (\frac{4 3 ^{0.86} \cdot 7 ^{0.14}}{25}) - 1)

Wende das Distributivgesetz an:

- (a - b) = - a + b

- (lo g_{2} (\frac{4 3 ^{0.86} \cdot 7 ^{0.14}}{25}) - 1) = - lo g_{2} (\frac{4 3 ^{0.86} \cdot 7 ^{0.14}}{25}) + 1

= - lo g_{2} (\frac{4 3 ^{0.86} \cdot 7 ^{0.14}}{25}) + 1

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